"a way of taming uncertainty, of turning raw data into arguments that can resolve profound questions" (Amabile 1989)

Success depends on evaluating the relative support for different models

Do long-winged crickets have a higher resting metabolic rate than short-winged crickets?

Controlling for body mass,

• Metabolic rates do not differ
• Average metabolic rate of long-wings is higher
• Average metabolic rate of long-wings is lower

Which set of loci best predicts oil content in corn kernels?

• Heritability of corn oil content is 0
• Different possible combinations of SNPs
• All additive
• Dominance
• Epistasis
• Structural variants

What are the phylogenetic relationships between dog breeds & other canids?

Is this bone more like a chimpanzee, australopith, Neandertal, or modern human?

• Relative probabilities of new fossil being in one of four groups
• What is the percent correct classification of known fossils?

Do alligators exhibit determinate or indeterminate growth and does growth depend of atmospheric oxygen levels?

Determinate:

• von Bertalanffy
• Logistic

Indeterminate:

• Polynomial

von Bertalanffy:

\[Femoral~Length\left(Age\right) = \theta_1 \left(1 - e^{-\theta_2 \left(Age - \theta_3\right)}\right)\]

Logistic:

\[Femoral~Length\left(Age\right) = \frac{\theta_1}{1 + e^{\frac{\theta_2 - Age}{\theta_3}}}\]

Polynomial:

\[Femoral~Length\left(Age\right) = a + b \cdot Age + c \cdot Age^2\]

Your Question Here

• Model 1
• Model 2
• Model 3
• …

1. Analytical
2. Maximum likelihood
3. Resampling
4. Bayesian

Algebraic (closed form) solutions for estimating parameters

\[\bar{Y} = \frac{\sum_{i=1}^{n}Y_i}{n}\]

• Historically, calculations had to be made by hand
• We made and continue to make a lot of strong inference from these calculations
  • The first ~75-100 years of experimental life sciences

Like the slope in bivariate regression, many are based of sums of squares (summed squared deviations).

\[b = \frac{\sum\left(X_{i}-\bar{X}\right)\left(Y_{i}-\bar{Y}\right)}{\sum\left(X_{i}-\bar{X}\right)^{2}}\]

For a given mean, how likely is observing my data?

• Maximum likelihood searches for the value of the mean (or other parameter of interest) that maximizes this likelihood.

Analytical solutions are generally equal to the maximum likelihood solutions (for set of assumptions).

• Few constraints on data and assumptions on tests
• Repeatedly sample from or shuffle the data many times, calculating a test statistic of interest each time
• "Monte Carlo" methods

What is the probability of these parameter estimates given the data and my prior knowledge?

• Inferences from data conditioned on prior knowledge
• Prior knowledge may be strong or weak

\[Pr[\theta | Data] = \frac{Pr[Data | \theta] \times Pr[\theta]}{Pr[Data]}\]

No quiz for this lecture.

Move on to Lecture 04-2.